May 10 U-Substitution

Basic idea

U-substitution is the reverse of the derivative chain rule. This is important when integrating an expression while chain rule is important while differentiating.

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Let's say you have the expression $$\dfrac{d}{dx} e^{\sin(x)} =\cos(x)e^{\sin(x)}$$. We know this because of the chain rule. Now, let's go backward.
$$e^{\sin(x)} = \int \cos(x)e^{\sin(x)}\, \mathrm{d}x$$
In order to solve this, we have to use the opposite of the chain rule: u-substitution.

Notice that $\cos(x)$ is the derivative of $\sin(x)$. We can replace $\sin(x)$ with $u(x)$.
$$\int u'(x) e^{u(x)}\, \mathrm{d}x$$
We rearrange to write this:
$$\int e^{u(x)}\, \boxed{u'(x)\mathrm{d}x}$$
Because it's the opposite of the chain rule, this equals $$e^{u(x)} + C = e^{\sin(x)} + C $$